On the analysis of ”simple” 2D stochastic cellular automata

Damien Regnault, Nicolas Schabanel, Eric Thierry

Abstract

Cellular automata are usually associated with synchronous
deterministic dynamics, and their asynchronous or stochastic
versions have been far less studied although significant for
modeling purposes. This paper analyzes the dynamics of a
two-dimensional cellular automaton, 2D Minority, for the Moore
neighborhood (eight closest neighbors of each cell) under fully
asynchronous dynamics (where one single random cell updates at each
time step). 2D Minority may appear as a simple rule, but it is known
from the experience of Ising models and Hopfield nets that 2D models
with negative feedback are hard to study. This automaton actually
presents a rich variety of behaviors, even more complex that what
has been observed and analyzed in a previous work on 2D Minority for
the von Neumann neighborhood (four neighbors to each cell)
(2007). This paper confirms the relevance of the later approach
(definition of energy functions and identification of competing
regions). Switching to the Moore neighborhood however strongly
complicates the description of intermediate configurations. New
phenomena appear (particles, wider range of stable
configurations). Nevertheless our methods allow to analyze different
stages of the dynamics. It suggests that predicting the behavior of
this automaton although difficult is possible, opening the way to
the analysis of the whole class of totalistic automata.